Group theory MOC

Conjugation by an element

In a group , given elements , we may conjugate by to get . Sometimes this is written as or , or for the right-action variant, .

Conjugation as an action

Conjugation by a given element is an automorphism of the group, such that constitutes a group action. The orbit of an element is its Conjugacy class , while its Stabilizer group is its centralizer group. An automorphism given by conjugation is called an inner automorphism, and the image forms the inner automorphism group.

Conjugacy relation

Given two group elements , we say is conjugate to () iff there exists such that . group The conjugacy relation is an Equivalence relation. group

Proof of equivalence relation

For any , , thus is reflexive. For any , , thus is symmetric. For any such that and , there exist such that and . Then and hence , wherefore is transitive. Therefore is an equivalence relation.

A conjugacy relation may also be applied between subgroups, see Conjugate subgroups.

Conjugacy class

The equivalence classes for the conjugacy relation form so-called conjugacy classes.

Properties

See also Inner group automorphism.

• since for all .
• iff and commute
• From above it follows that in an Abelian group all conjugacy classes are singletons.
• A conjugacy class is not necessarily a subgroup (since it is either the trivial subgroup or ).
• By the Orbit-stabilizer theorem, .
• The number of conjugacy classes equals the number of non-equivalent irreps of a group.

Examples

• In rotations by the same angle (i.e. only differing in axis of rotation) form conjugacy classes.
• In elements are conjugate to each other iff they have similar matrices (in subgroups, such as , conjugacy may be more restricted, however all conjugate elements are similar).

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